# Regularity of symbolic powers and Arboricity of matroids

## TL;DR

This paper computes the regularity of symbolic powers of Stanley-Reisner ideals associated with matroids, linking combinatorial properties like arboricity and circumference of dual matroids.

## Contribution

It provides explicit formulas for regularity of symbolic powers of matroid Stanley-Reisner ideals and establishes a sharp bound between arboricity and circumference of dual matroids.

## Key findings

- Explicit regularity formulas for symbolic powers of matroid ideals
- A sharp bound relating arboricity and circumference of dual matroids
- Connections between combinatorial matroid properties and algebraic invariants

## Abstract

Let $\Delta$ be a simplicial complex of a matroid $M$. In this paper, we explicitly compute the regularity of all the symbolic powers of a Stanley-Reisner ideal $I_\Delta$ in terms of combinatorial data of the matroid $M$. In order to do that, we provide a sharp bound between the arboricity of $M$ and the circumference of its dual $M^*$.